HFixed-lag smoothing for discrete linear time-varying systems

Huanshui Zhang, Lihua Xie, Yeng Chai Soh, David Zhang

Abstract: This paper is concerned with the finite horizon H∞<math><msub is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mo is="true">∞</mo></mrow></msub></math> fixed-lag smoothing problem for discrete linear time-varying systems. The existence of an H∞<math><msub is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mo is="true">∞</mo></mrow></msub></math> smoother is first related to certain inertia condition of an innovation matrix. The innovation matrix is traditionally computed via a Riccati difference equation (RDE) associated with the H∞<math><msub is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mo is="true">∞</mo></mrow></msub></math> filtering of an augmented system which is computationally expensive. To avoid solving the RDE of high dimension, we introduce a re-organized innovation and apply innovation analysis and projection theory in Krein space to give a simple method of computing the innovation matrix. The H∞<math><msub is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mo is="true">∞</mo></mrow></msub></math> smoother is computed as a projection in Krein space by performing two RDEs of the same dimension as that of the original system.